Note on Twisted Elliptic Genus of K3 Surface

TL;DR

This work tests the hypothesis that the Mathieu group $M_{24}$ acts as a symmetry on the K3 surface by constructing all twisted elliptic genera $Z_g(z;\tau)$ for its 26 conjugacy classes (types I and II) and using $M_{24}$-character theory to decompose the K3 elliptic genus into irreducible representations. Type I twists were previously known, while type II twists are newly derived as eta-quotients times $\phi_{-2,1}$, with modularity properties tied to $\mathrm{ord}(g)$. The authors show that the resulting multiplicities $c_R(n)$ are positive integers up to $n=600$, providing strong evidence for Mathieu moonshine, and discuss entropy reductions in twisted sectors consistent with $\mathrm{ord}(g)$. Overall, the paper advances understanding of hidden $M_{24}$ symmetry in string theory on K3 and links twisted elliptic genera to moonshine phenomena. The results suggest a rich, structure-rich action of $M_{24}$ on the K3 state space and offer concrete computational data for further exploration of underlying geometric and algebraic origins.

Abstract

We discuss the possibility of Mathieu group M24 acting as symmetry group on the K3 elliptic genus as proposed recently by Ooguri, Tachikawa and one of the present authors. One way of testing this proposal is to derive the twisted elliptic genera for all conjugacy classes of M24 so that we can determine the unique decomposition of expansion coefficients of K3 elliptic genus into irreducible representations of M24. In this paper we obtain all the hitherto unknown twisted elliptic genera and find a strong evidence of Mathieu moonshine.